Question

When only the front wheels of an automobile are on a platform scale, the scale balances at \(8.00 \mathrm{kN}\); when only the rear wheels are on the scale, it balances at \(6.00 \mathrm{kN}\). What is the weight of the automobile, and how far is its center of mass behind the front axle? The distance between the axles is \(2.80 \mathrm{~m}\).

Short answer

Answer: The weight of the automobile is 14.00 kN, and the center of mass is 1.20 m behind the front axle.

Step-by-step solution

Convert given information to variables

Let's define the variables for given information: - Front wheel force, \(F_f = 8.00 \mathrm{kN}\) - Rear wheel force, \(F_r = 6.00 \mathrm{kN}\) - Distance between axles, \(d = 2.80 \mathrm{m}\)

Calculate total weight of the automobile

The total weight of the automobile, \(W\), is the sum of the forces exerted by the front and rear wheels. Thus, we can calculate it as follows: \(W = F_f + F_r = 8.00 \mathrm{kN} + 6.00 \mathrm{kN} = 14.00 \mathrm{kN}\)

Calculate the position of the center of mass

Now we have to find the position of the center of mass relative to the front axle, which we will denote as \(x\). We can solve for this variable by setting up a moment equilibrium equation about the front axle: \(\sum M_{front \, axel} = 0\) \(F_r \cdot d - W \cdot x = 0\) To find the position of the center of mass, \(x\), we can rearrange the equation: \(x = \frac{F_r \cdot d}{W}\)

Substitute given values and calculate x

Now we can substitute the given values into the equation we found in Step 3 to find the position of the center of mass \(x\): \(x = \frac{6.00 \mathrm{kN} \cdot 2.80 \mathrm{m}}{14.00 \mathrm{kN}}\) \(x = 1.20 \mathrm{m}\) So the position of the center of mass is \(1.20 \mathrm{m}\) behind the front axle. In summary, the weight of the automobile is \(14.00 \mathrm{kN}\), and the center of mass is \(1.20 \mathrm{m}\) behind the front axle.

Key concepts

Center of Mass

Understanding where the center of mass of an object is located is crucial for analyzing the balance and stability of that object. In the case of an automobile, the center of mass is the point where the car's mass is evenly distributed in all directions. The closer this point is to the ground, the more stable the automobile is, especially during turns or when quick maneuvers are required.

When measuring the weight on the platform scale with only one set of wheels, you gather information on how the car's weight is distributed. By doing this for the front and rear wheels, you can deduce the center of mass. If the weight reading from the front is higher than from the back, it indicates that the center of mass is closer to the front axle, which is a common design in front-engine cars.

Moment Equilibrium

Moment equilibrium is a principle derived from the physics concept of torque. When an object is in equilibrium, the sum of all moments (or torques) around any axis is zero. In automotive terms, this means the vehicle will not rotate around any axle, indicating a stable weight distribution.

Applying this concept to the textbook problem allows us to calculate the automobile’s total weight and, importantly, the location of its center of mass by setting up an equilibrium around the front axle. The moments created by the force on the rear wheels and the total weight of the car must balance for the car to remain stable. This balance is essential for ensuring the vehicle's handling characteristics are predictable and safe.

Physics Problem Solving

Physics problem-solving involves using theoretical frameworks and mathematical tools to understand and predict the physical world. A systematic approach, similar to the one used in the automobile weight distribution problem, can often be applied:

• Define the variables
• Apply physical laws, like conservation of energy or moment equilibrium
• Set up relevant equations
• Solve mathematically

This structured method helps to break down complex scenarios into manageable parts, ensuring that all factors are considered and the solution is thorough. It is an invaluable skill not only in physics but across many disciplines that require analytical thinking.

Force Measurement

Force measurement is a fundamental aspect of physics that involves determining the magnitude of forces acting on an object. In the context of an automobile, as presented in the exercise, force measurements are taken using a platform scale to weigh the forces exerted by the front and rear wheels. This process is crucial to determining vehicle weight, tire pressure requirements, and optimizing weight distribution for performance and safety.

Knowing the individual weights on each set of wheels leads us to the vehicle's total weight and helps in calculating the center of mass. With modern vehicles becoming more advanced, accurate force measurement is vital for adaptive suspension systems that adjust the ride dynamically based on the vehicle load and driving conditions.